teaching/scientificComputing
Archived
3
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

[pointprocesses] improved chapter and added rasterplot()

Browse Source
This commit is contained in:
Jan Benda
2018-11-26 19:05:57 +01:00
parent c72ea6ad37
commit acf7fc8c81
10 changed files with 104 additions and 83 deletions
Download Patch File Download Diff File Expand all files Collapse all files

2
pointprocesses/lecture/pointprocesses-chapter.tex
View File

@@ -18,7 +18,7 @@
\section{TODO}
\begin{itemize}
\item Add spikeraster function
\item Explain difference stationary versus non-stationary point process
\item Multitrial firing rates
\item Choice of bin width for PSTH, kernel width, also in relation sto
stimulus time scale

160
pointprocesses/lecture/pointprocesses.tex
View File

@@ -2,17 +2,15 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Spiketrain analysis}
\selectlanguage{english}
\enterm[Action potentials]{action potentials} (\enterm{spikes}) are
the carriers of information in the nervous system. Thereby it is the
time at which the spikes are generated that is of importance for
information transmission. The waveform of the action potential is
largely stereotyped and therefore does not carry information.
\enterm[Actionspotentials]{Actionspotentials} (\enterm{spikes}) are
the carriers of information in the nervous system. Thereby it is
mainly the time at which the spikes are generated that is of
importance. The waveform of the action potential is largely
stereotyped and does not carry information.
The result of the processing of electrophysiological recordings are
series of spike times, which are then termed \enterm{spiketrains}. If
measurements are repeated we yield several \enterm{trials} of
The result of the pre-processing of electrophysiological recordings are
series of spike times, which are termed \enterm{spiketrains}. If
measurements are repeated we get several \enterm{trials} of
spiketrains (\figref{rasterexamplesfig}).
Spiketrains are times of events, the action potentials. The analysis
@@ -21,34 +19,32 @@ of these leads into the realm of the so called \enterm[point
\begin{figure}[ht]
\includegraphics[width=1\textwidth]{rasterexamples}
\titlecaption{\label{rasterexamplesfig}Raster-plot.}{Raster-plot of
ten realizations of a stationary point process (homogeneous point
process with a rate $\lambda=20$\;Hz, left) and an inhomogeneous
point process (perfect integrate-and-fire neuron dirven by
Ohrnstein-Uhlenbeck noise with a time-constant $\tau=100$\,ms,
right). Each vertical dash illustrates the time at which the
action potential was observed. Each line represents the event of
each trial.}
\titlecaption{\label{rasterexamplesfig}Raster-plot.}{Raster-plots of
ten trials of data illustrating the times of the action
potentials. Each vertical dash illustrates the time at which an
action potential was observed. Each line displays the events of
one trial. Shown is a stationary point process (left, homogeneous
point process with a rate $\lambda=20$\;Hz, left) and an
non-stationary point process (right, perfect integrate-and-fire
neuron dirven by Ohrnstein-Uhlenbeck noise with a time-constant
$\tau=100$\,ms, right).}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Point processes}
A temporal \enterm{point process} is a stochastic process that
generates a sequence of events at times $\{t_i\}$, $t_i \in
\reZ$.
\begin{ibox}{Examples of point processes}
Every point process is generated by a temporally continuously
developing process. An event is generated whenever this process
reaches a certain threshold. For example:
crosses some threshold. For example:\vspace{-1ex}
\begin{itemize}
\item Action potentials/heart beat: created by the dynamics of the
neuron/sinoatrial node
\item Earthquake: defined by the dynamics of the pressure between
tectonical plates.
\item Evoked communication calls in crickets/frogs/birds: shaped by
the dynamics of nervous system and the muscle appartus.
\item Communication calls in crickets/frogs/birds: shaped by
the dynamics of the nervous system and the muscle appartus.
\end{itemize}
\end{ibox}
@@ -60,43 +56,51 @@ generates a sequence of events at times $\{t_i\}$, $t_i \in
$T_i=t_{i+1}-t_i$ and the number of events $n_i$. }
\end{figure}
In the neurosciences, the statistics of point processes is of
importance since the timing of the neuronal events (the action
potentials) is crucial for information transmission and can be treated
as such a process.
A temporal \enterm{point process} is a stochastic process that
generates a sequence of events at times $\{t_i\}$, $t_i \in \reZ$. In
the neurosciences, the statistics of point processes is of importance
since the timing of neuronal events (action potentials, post-synaptic
potentials, events in EEG or local-field recordings, etc.) is crucial
for information transmission and can be treated as such a process.
Point processes can be described using the intervals between
successive events $T_i=t_{i+1}-t_i$ and the number of observed events
within a certain time window $n_i$ (\figref{pointprocessscetchfig}).
The events of a point process can be illustrated by means of a raster
plot in which each vertical line indicates the time of an event. The
event from two different point processes are shown in
\figref{rasterexamplesfig}. Point processes can be described using
the intervals between successive events $T_i=t_{i+1}-t_i$ and the
number of observed events within a certain time window $n_i$
(\figref{pointprocessscetchfig}).
The events originating from a point process can be illustrated in form
of a scatter- or raster plot in which each vertical line indicates the
time of an event. The event from two different point processes are
shown in \figref{rasterexamplesfig}.
\begin{exercise}{rasterplot.m}{}
Implement a function \code{rasterplot()} that displays the times of
action potentials within the first \code{tmax} seconds in a raster
plot. The spike times (in seconds) recorded in the individual trials
are stored as vectors of times within a \codeterm{cell-array}.
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Intervalstatistics}
\section{Interval statistics}
The intervals $T_i=t_{i+1}-t_i$ between successive events are real
positive numbers. In the context of action potentials they are
referred to as \enterm{interspike intervals}. The statistics of these
are described using the common measures.
referred to as \enterm{interspike intervals}. The statistics of
interspike intervals are described using common measures for
describing the statistics of stochastic real-valued variables:
\begin{figure}[t]
\includegraphics[width=0.96\textwidth]{isihexamples}\vspace{-2ex}
\titlecaption{\label{isihexamplesfig}Interspike interval
histogram}{of the spikes depicted in \figref{rasterexamplesfig}.}
histograms}{of the spike trains shown in \figref{rasterexamplesfig}.}
\end{figure}
\begin{exercise}{isis.m}{}
Implement a function \code{isis()} that calculates the interspike
intervals from several spike trains. The function should return a
single vector of intervals. The action potentials recorded in the
individual trials are stored as vectors of spike times within a
\codeterm{cell-array}. Spike times are given in seconds.
single vector of intervals. The spike times (in seconds) of each
trial are stored as vectors within a \codeterm{cell-array}.
\end{exercise}
\subsection{First order interval statistics}
%\subsection{First order interval statistics}
\begin{itemize}
\item Probability density $p(T)$ of the intervals $T$
(\figref{isihexamplesfig}). Normalized to $\int_0^{\infty} p(T) \; dT
@@ -138,10 +142,17 @@ and non-stationary return maps are distinctly different
\begin{figure}[t]
\includegraphics[width=1\textwidth]{returnmapexamples}
\includegraphics[width=1\textwidth]{serialcorrexamples}
\titlecaption{\label{returnmapfig}Interspike interval analyses of a
stationary and a non-stationary pointprocess.}{Upper plots show the
return maps and the lower panels depict the serial correlation of
successive intervals separated by the lag $k$.}
\titlecaption{\label{returnmapfig}Interspike interval
correlations}{of the spike trains shown in
\figref{rasterexamplesfig}. Upper panels show the return maps and
lower panels the serial correlations of successive intervals
separated by lag $k$. All the interspike intervals of the
stationary spike trains are independent of each other --- this is
a so called \enterm{renewal process}. In contrast, the ones of the
non-stationary spike trains show positive correlations that decay
for larger lags. The positive correlations in this example are
caused by a common stimulus that slowly increases and decreases
the mean firing rate of the spike trains.}
\end{figure}
Such dependencies can be further quantified using the \enterm{serial
@@ -170,17 +181,18 @@ with itself and is always 1.
% \includegraphics[width=0.48\textwidth]{poissoncounthist100hz100ms}
% \titlecaption{\label{countstatsfig}Count Statistik.}{}
% \end{figure}
The number of events $n_i$ (counts) in a time window $i$ of the duration $W$
Counting the number of events $n_i$ (counts) in time windows $i$ of duration $W$
yields positive integer random numbers that are commonly quantified
using the following measures:
\begin{itemize}
\item Histogram of the counts $n_i$.
\item Average number of events: $\mu_N = \langle n \rangle$.
\item Variance of the counts: $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
\item \determ{Fano Faktor} (The variance divided by the average): $F = \frac{\sigma_n^2}{\mu_n}$.
\item Average number of counts: $\mu_n = \langle n \rangle$.
\item Variance of counts: $\sigma_n^2 = \langle (n - \langle n \rangle)^2 \rangle$.
\item \determ{Fano Factor} (variance of counts divided by average count): $F = \frac{\sigma_n^2}{\mu_n}$.
\end{itemize}
And in particular the average firing rate $r$ (spike count per time interval
, \determ{Feuerrate}) that is given in Hertz \sindex[term]{Feuerrate!mittlere Rate}
Of particular interest is the average firing rate $r$ (spike count per
time interval , \determ{Feuerrate}) that is given in Hertz
\sindex[term]{firing rate!average rate}
\begin{equation}
\label{firingrate}
r = \frac{\langle n \rangle}{W} \; .
@@ -204,30 +216,30 @@ And in particular the average firing rate $r$ (spike count per time interval
the distribution of spike counts observed in a certain time
window. The function should take two input arguments: (i) a
\codeterm{cell-array} of vectors containing the spike times in
seconds observed in a number of trials and (ii) the duration of the
time window that is used to evaluate the counts.
seconds observed in a number of trials, and (ii) the duration of the
time window that is used to evaluate the counts.\pagebreak[4]
\end{exercise}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Homogeneous Poisson process}
The Gaussian distribution is, due to the central limit theorem, the
standard for continuous measures. The equivalent in the realm of point
processes is the \enterm{Poisson distribution}.
The Gaussian distribution is, because of the central limit theorem,
the standard distribution for continuous measures. The equivalent in
the realm of point processes is the \enterm{Poisson distribution}.
In a \enterm[Poisson process!homogeneous]{homogeneous Poisson process}
the events occur at a fixed rate $\lambda=\text{const.}$ and are
the events occur at a fixed rate $\lambda=\text{const}$ and are
independent of both the time $t$ and occurrence of previous events
(\figref{hompoissonfig}). The probability of observing an even within a
small time window of width $\Delta t$ is given by
(\figref{hompoissonfig}). The probability of observing an event within
a small time window of width $\Delta t$ is given by
\begin{equation}
\label{hompoissonprob}
P = \lambda \cdot \Delta t \; .
\end{equation}
In an \enterm[Poisson process!inhomogeneous]{inhomogeneous Poisson
process}, however, the rate $\lambda$ depends on the time: $\lambda =
process}, however, the rate $\lambda$ depends on time: $\lambda =
\lambda(t)$.
\begin{exercise}{poissonspikes.m}{}
@@ -249,7 +261,11 @@ In an \enterm[Poisson process!inhomogeneous]{inhomogeneous Poisson
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{poissonisihexp20hz}\hfill
\includegraphics[width=0.45\textwidth]{poissonisihexp100hz}
\titlecaption{\label{hompoissonisihfig}Distribution of interspike intervals of two Poisson processes.}{}
\titlecaption{\label{hompoissonisihfig}Distribution of interspike
intervals of two Poisson processes.}{The processes differ in their
rate (left: $\lambda=20$\,Hz, right: $\lambda=100$\,Hz). The red
lines indicate the corresponding exponential interval distribution
\eqnref{poissonintervals}.}
\end{figure}
The homogeneous Poisson process has the following properties:
@@ -267,7 +283,10 @@ The homogeneous Poisson process has the following properties:
process is also called a \enterm{renewal process}.
\item The number of events $k$ within a temporal window of duration
$W$ is Poisson distributed:
\[ P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!} \]
\begin{equation}
\label{poissoncounts}
P(k) = \frac{(\lambda W)^ke^{\lambda W}}{k!}
\end{equation}
(\figref{hompoissoncountfig})
\item The Fano Faktor is always $F=1$ .
\end{itemize}
@@ -286,8 +305,11 @@ The homogeneous Poisson process has the following properties:
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz10ms}\hfill
\includegraphics[width=0.48\textwidth]{poissoncounthistdist100hz100ms}
\titlecaption{\label{hompoissoncountfig}Count statistics of Poisson
spiketrains.}{}
\titlecaption{\label{hompoissoncountfig}Distribution of counts of a
Poisson spiketrain.}{The count statistics was generated for two
different windows of width $W=10$\,ms (left) and width $W=100$\,ms
(right). The red line illustrates the corresponding Poisson
distribution \eqnref{poissoncounts}.}
\end{figure}
@@ -312,7 +334,7 @@ closely.
\begin{figure}[tp]
\includegraphics[width=\columnwidth]{firingrates}
\titlecaption{Estimating the time-dependent firing
rate.}{\textbf{A)} Rasterplot depicting the spiking activity of a
rate.}{\textbf{A)} Rasterplot showing the spiking activity of a
neuron. \textbf{B)} Firing rate calculated from the
\emph{instantaneous rate}. \textbf{C)} classical PSTH with the
\emph{binning} method. \textbf{D)} Firing rate estimated by
@@ -529,5 +551,3 @@ kernel.
the same size as the original stimulus contained in file
\file{sta\_data.mat}.
\end{exercise}
\selectlanguage{english}

2
pointprocesses/lecture/pointprocessscetch.gpt
View File

@@ -1,4 +1,4 @@
set term epslatex size 11.4cm, 6.5cm
set term epslatex size 11.4cm, 6.4cm
set out 'pointprocessscetch.tex'
set border 0

4
pointprocesses/lecture/pointprocessscetchA.eps
View File

@@ -1,7 +1,7 @@
%!PS-Adobe-2.0 EPSF-2.0
%%Title: pointprocessscetchA.tex
%%Creator: gnuplot 4.6 patchlevel 4
%%CreationDate: Tue Nov 28 10:02:49 2017
%%CreationDate: Mon Nov 26 17:57:49 2018
%%DocumentFonts:
%%BoundingBox: 50 50 373 135
%%EndComments
@@ -433,7 +433,7 @@ SDict begin [
/Author (benda)
% /Producer (gnuplot)
% /Keywords ()
/CreationDate (Tue Nov 28 10:02:49 2017)
/CreationDate (Mon Nov 26 17:57:49 2018)
/DOCINFO pdfmark
end
} ifelse

BIN
pointprocesses/lecture/pointprocessscetchA.pdf
View File

Binary file not shown.

4
pointprocesses/lecture/pointprocessscetchB.eps
View File

@@ -1,7 +1,7 @@
%!PS-Adobe-2.0 EPSF-2.0
%%Title: pointprocessscetchB.tex
%%Creator: gnuplot 4.6 patchlevel 4
%%CreationDate: Tue Nov 28 10:02:49 2017
%%CreationDate: Mon Nov 26 17:57:49 2018
%%DocumentFonts:
%%BoundingBox: 50 50 373 237
%%EndComments
@@ -433,7 +433,7 @@ SDict begin [
/Author (benda)
% /Producer (gnuplot)
% /Keywords ()
/CreationDate (Tue Nov 28 10:02:49 2017)
/CreationDate (Mon Nov 26 17:57:49 2018)
/DOCINFO pdfmark
end
} ifelse

BIN
pointprocesses/lecture/pointprocessscetchB.pdf
View File

Binary file not shown.